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I'm little bit confused, when I want to standardize a normal distribution, when to use standard error and when to use standard deviation ?. In statistics we often use samples so, is it always we use standard error to estimate the population parameters and so, or What ?.

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Incidentally, this exact question came up with a colleague today.

If you want to do the z-score, you use standard deviation: $z = \frac{x - \bar{x}}{s}$. Standard deviation is a property of a population, and we use some formula (often $s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2)$ to estimate the population parameter. The population parameter does not depend on the sample size or even the sample at all.

Standard error has to do with the so-called sampling distribution of a statistic. This does depend on the sample size. Standard error of the mean is what most people see first. You know the formula: $\dfrac{s}{\sqrt{n}}$.

The sampling distribution of a statistic follows the following logic.

1) Pick of statistic of interest, say $\bar{x}$.

2) Grab a sample of size n.

3) Calculate $\bar{x}$.

4) Repeat 2 and 3 (infinitely) many times.

5) Plot the values of $\bar{x}$ that you calculated (something like a histogram). to get the sampling distribution.

Since you should have tighter and tighter estimates of $\mu$ as you get larger and larger sample sizes, your $\bar{x}$ values should be very close to $\mu$ when you have a lot of data, and this is why standard error shrinks as you increase the sample size. However, if we sampled from a standard normal population, no matter how big the sample size (and correspondingly small the standard error), the population has a standard deviation of 1.

Since students first learn about standard error of the mean, which explicitly depends on the population (or estimated) standard deviation, and students often do not progress past testing means, it's easy to confuse the two. (In fact, the standard error is the standard deviation of the sampling distribution.) However, you can find a standard error for other statistics. Look at those steps 1-5. You could repeat that process for a different statistic, perhaps $s^2$, median, or IQR. If you know how to write loops in a language like Python or R (or whatever statistical language), I'd encourage you to try writing a simulation of these.

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  • $\begingroup$ So if I had an sample taken from a population and I want to get a Confidence Interval of population mean or even perform a Hypothesis test on population mean I will use standard error !? $\endgroup$ Commented Aug 7, 2019 at 0:13
  • $\begingroup$ @AbdEl-RahmanAkram Exactly, because your confidence interval should shrink as you gather more data, and you should be able to reject a null hypothesis with less difference as you gather more data. $\endgroup$
    – Dave
    Commented Aug 7, 2019 at 0:15
  • $\begingroup$ I get it know thanks for help. $\endgroup$ Commented Aug 7, 2019 at 0:18
  • $\begingroup$ @AbdEl-RahmanAkram I'm glad I could help! If you could click up up-vote, I'd appreciate it. :) $\endgroup$
    – Dave
    Commented Aug 7, 2019 at 0:19
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    $\begingroup$ the best explanation ever! $\endgroup$
    – daniel_hck
    Commented Mar 29, 2021 at 17:43

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